Tangent to an Offset Circle at a Point

The radius runs from the center (1, −2) to (4, −1), a direction of (3, 1), so its slope is 1/3. The tangent is perpendicular, with slope −3: y + 1 = −3(x − 4), which simplifies to 3x + y = 11. Check: 3(4) + (−1) = 11. The line x − 3y = 7 also passes through the point but runs along the radius, not across it. TI-36X Pro tip: two choices pass through (4, −1), so test the slope too. The radius slope is (−1 − (−2)) ÷ (4 − 1) = 1/3, and a tangent must have slope −3, which matches 3x + y = 11 (y = −3x + 11).